The document provides a comprehensive overview of the properties of exponents, including definitions and examples for positive, negative, and zero exponents, as well as rules for exponent operations. It covers specific properties such as the product of powers, power of a power, and root functions in relation to exponents. Additionally, it explains how to handle base cases like 0, 1, and -1 raised to various powers.

1. ENZO EXPOSYTO
MATHS
SYMBOLS
PROPERTIES of EXPONENTS
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2. 23 22 21 20
EXPONENTS
2-1 2-2 2-3 2-4
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3.
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4. 1 - Exponents and Their Properties - 1 5
2 - Exponents and Their Properties - 2 10
3 - Exponents and Their Properties - 3 13
4 - Exponents and Their Properties - 4 15
5 - SitoGraphy 19
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5. EXPONENTS
and THEIR
PROPERTIES
(1)
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6. a superscript n
an say: a superscript n;
(a^n) a is a positive number and n is a positive integer;
The base a is raised to the power of n;
an is equal to
(we write ‘a’, n times
and
the multiplication sign ‘×’, n-1 times):
an = a × a × ... × a
for example:
23 = 2 × 2 × 2 = 8
(3 times "2" and 3-1 times "x")
a1 say: a superscript 1
(a^1) It’s equal to a: a1= a;
for example:
51 = 5
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7. a superscript n super superscript m
say: a superscript n super superscript m
a is a positive number; n and m are positive integers;
a^n^m It’s equal to a^(nm):
a^n^m = a^(nm);
for example:
2^3^2 = 2^(32) = 29 = 512
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8. 0 superscript n AND 0 superscript (-n)
0n n is a positive integer;
(0^n) it’s equal to 0:
0n = 0;
for example:
05 = 0 × 0 × 0 × 0 × 0 = 0
0-n n is a positive integer;
(0^(-n)) 0-n is impossible
and the result Does Not Exist (DNE)
for example:
0-1 = 1 = 1 is impossible (it’s impossible dividing 1 by 0)
01 0
and the result Does Not Exist (DNE)
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9. 1 superscript n AND (-1) superscript n
1n It is equal to 1:
(1^n) 1n = 1;
for example:
15 = 1 × 1 × 1 × 1 × 1 = 1
(-1)n n even;
it is equal to +1;
for example:
(-1)4 = (-1) × (-1) × (-1) × (-1) = +1
(-1)n n odd;
it is equal to -1;
for example
(-1)5 = (-1) × (-1) × (-1) × (-1) × (-1) = -1
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10. EXPONENTS
and THEIR
PROPERTIES
(2)
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11. Exponents - 6 Properties - 2A
EXPONENTS and THEIR PROPERTIES - 2A
[a, b elements of R+]
[m, n elements of Z+]
Z+ = {1, 2, 3, …}
Property Powers Exponents Powers Exponents Result
1st am · an = am + n 23 · 22 = 23 + 2 = 32
2nd am
=
an
am -n
23
=
22
23 -2 = 2
3rd (am)n = am *n (23)2 = 23 *2 = 64
4th n√am = am : n 2√24 = 24 : 2 = 4
5th an · bn = (a · b)n 22 · 32 = (2 · 3)2 = 36
6th
an
———- =
bn
(_a_)n
b
43
———- =
23
(_4_)3
=
2
8
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12. Exponents - 6 Properties - 2B
EXPONENTS and THEIR PROPERTIES - 2B
[a, b elements of R+]
[m, n elements of Z+]
Z+ = {1, 2, 3, …}
Property Exponents Powers Exponents Powers Result
1st am + n = am · an
23 + 2 = 23 · 22 = 32
2nd
am -n = am
an
23 -2 = 23
=
22
2
3rd am *n = (am)n
23 *2 = (23)2 = 64
4th am : n = n√am 24 : 2 = 2√24 = 4
5th (a · b)n = an · bn (2 · 3)2 = 22 · 32 = 36
6th
(_a_)n =
b
an
———-
bn
(_4_)3
=
2
43
———- =
23
8
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13. EXPONENTS
and THEIR
PROPERTIES
(3)
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14. Exponents - 6 Properties - Proofs/Examples
PROOFS / EXAMPLES
[a, b elements of R+]
[the exponents m and n are elements of Z+]
Z+ = {1, 2, 3, …}
1st a3 · a2 = (a · a · a) · (a · a) = a · a · a · a · a = a5 = a3 + 2
2nd a3 = (a · a · a) = a = a1 = a3 - 2
a2 (a · a)
3rd (a3)2 = (a · a · a) · (a · a · a) = a · a · a · a · a · a = a6 = a3 * 2
4th 2√a4 = 2√(a · a · a · a) = a · a = a2 = a4 : 2
5th a3 · b3 = (a · a · a) · (b · b· b) = a · b · a · b · a · b = … = (a·b)3
6th
a3 (a · a · a) a a a a
——- = ————— = —— . —— . —— = … = (—)3
b3 (b · b· b) b b b b
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15. EXPONENTS
and THEIR
PROPERTIES
(4)
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16. Exponents - 2nd property- a superscript 0
EXPONENTS and THEIR PROPERTIES - 4A
Z+ = {1, 2, 3, …}
Reference Property Notice Proof Example
2nd
Property
a0 = 1
a ≠ 0
n Z+ a0 = an - n 20 = 23 - 3
= an
an
= 23
23
= 1 = 1
1 = a0 a ≠ 0 1 = 20
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17. Exponents - 2nd property - a superscript (-n)
EXPONENTS and THEIR PROPERTIES - 4B
Z+ = {1, 2, 3, …}
Reference Property Notice Proof Example
2nd
Property
a-n = 1
an
a ≠ 0
n Z+ a-n = a0-n 2-3 = 20-3
= a0
an
= 20
23
= 1
an
= 1
23
1 = a-n
an
a ≠ 0
n Z+
1 = 2-3
23
Negative exponents are
the reciprocals of the positive exponents:
an = 1
a-n
a ≠ 0
n Z+
23 = 1
2-3
1 = an
a-n
a ≠ 0
n Z+
1 = 23
2-3
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18. Exponents - 4th property - a superscript (1/n)
EXPONENTS and THEIR PROPERTIES - 4C
Z+ = {1, 2, 3, …}
Reference Property Notice Proof Example
4th
Property
n√a = a1/n
a ≥ 0
n Z+
n√a = n√a1 3√8 = 3√81
= a1/n = 81/3
a1/n = n√a
a ≥ 0
n Z+ 81/3 = 3√8
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19. SitoGraphy
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20. http://www.gh-mathspeak.com/examples/grammar-rules/?rule=scripts
http://www.rapidtables.com/math/number/exponent.htm
http://www.mathplanet.com/education/algebra-1/exponents-and-exponential-functions/properties-of-exponents
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